We will introduce two new Li-Yau estimates for the heat equation
on manifolds under some new curvature conditions. The first one is obtained
for n-dimensional manifolds with fixed Riemannian metric under the
condition that the Ricci curvature being L^p bounded for some p>n/2. The
second one is proved for manifolds evolving under the Ricci flow with
uniformly bounded scalar curvature. Moreover, we will also apply the first
Li-Yau estimate to generalize Colding-Naber's results on parabolic
approximations of local Busemann functions to weaker curvature condition
setting. This is a recent joint work with Richard Bamler and Qi S. Zhang.